The remainder of this paper is organized as follows. In Section 2, we discuss the behavior of Bayesian posterior distribution under a misspecified model and define and review the literature on generalized Bayes posteriors. For the latter, the choice of learning rate is essential, and we provide details for four recently proposed learning rate selection methods in Section 3. Then, in Section 4, we show a simple illustrative example to give some intuition about how the different methods perform and, in particular, this suggests that the methods which are not designed specifically to calibrate the credible region's coverage probability may not be able to achieve the nominal level in general. Simulation results are presented in Sections 5–6, for linear and binary regression models, and the take-away message is that the method of [1]} is more stable in achieving the coverage probability across different sample sizes and misspecification degrees compared to the others. Some concluding remarks are given in Section 7.
